Free-energy distribution of the directed polymer at high temperature

¹  Dipartimento di Fisica dell'Università di Pisa and INFN - 56127 Pisa, Italy, EU
²  CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure - 24 rue Lhomond, 75231 Paris Cedex, France, EU
³  CNRS-Université Paris-Sud, LPTMS, UMR8626 - Bât. 100, 91405 Orsay Cedex, France, EU

Corresponding author: ledou@lpt.ens.fr

Received: 24 February 2010
Accepted: 8 April 2010

Abstract

We study the directed polymer of length t in a random potential with fixed endpoints in dimension 1+1 in the continuum and on the square lattice, by analytical and numerical methods. The universal regime of high temperature T is described, upon scaling “time” t ~ T⁵/κ and space x = T³/κ (with κ = T for the discrete model) by a continuum model with δ-function disorder correlation. Using the Bethe Ansatz solution for the attractive boson problem, we obtain all positive integer moments of the partition function. The lowest cumulants of the free energy are predicted at small time and found in agreement with numerics. We then obtain the exact expression at any time for the generating function of the free-energy distribution, in terms of a Fredholm determinant. At large time we find that it crosses over to the Tracy-Widom distribution (TW) which describes the fixed-T infinite-t limit. The exact free-energy distribution is obtained for any time and compared with very recent results on growth and exclusion models.